Concluding remark. As we have already mentioned in Introduction, in practice, to check the solvability by generalized quadratures of a given linear differential system \({\rm d}y/{\rm d}z =B(z)y\) , it is sufficient (but might be very costly) to compute the Lie algebra of the corresponding differential Galois group (for example, by the algorithm in [1]}) and this works, in theory at least, for a general system. Now for the class of systems with small coefficients it is much better (from the computational point of view) to apply the method of the present paper, namely computing the Lie algebra defined by the coefficients of the matrix \(B(z)\) . Indeed, let us denote by \(\mathfrak {g} \subset {\mathfrak {gl}}_n({\mathbb {C}})\)  the Lie algebra of the differential Galois group of our system and by \(\mathfrak {h}\)  the Lie algebra generated by the coefficient matrix. The latter is defined as follows: if \(B(z) = f_1M_1 + \dots + f_sM_s\) , where the \(M_i\) 's are constant matrices and the \(f_i\) 's are rational functions in \({\mathbb {C}}(z)\)  that form a basis of the \(\mathbb {C}\) -vector space generated by the entries of \(B(z)\) , then \(\mathfrak {h}\)  is the matrix Lie algebra generated by the \(M_i\) 's. One has that \(\mathfrak {g}\)  is always a Lie subalgebra of \(\mathfrak {h}\) . In general this inclusion is strict, and when the equality holds the differential system is called in reduced form (see [1]} and references therein). Now, due to the Kolchin–Kovacić theorem, there exists a gauge transformation \(y = P \tilde{y}\) , with \(P\)  having its entries in the algebraic closure of \({\mathbb {C}}(z)\) , that takes the given system \({\rm d}y/{\rm d}z = B(z) y\)  into an equivalent one \({\rm d}\tilde{y}/{\rm d}z=\tilde{B}(z) \tilde{y}\)  that is in reduced form. The algorithm in [1]} computes such a transformation and hence gets \(\mathfrak {g}\)  as the Lie algebra defined by \(\tilde{B}(z)\) . However computing a reduced form is not an easy task at all, since this is equivalent to finding the differential Lie algebra \(\mathfrak {g}\) . Now, for systems with small coefficients what we propose is considering the set of the matrices \(B_i^{(j)}\)  involved in the partial fraction decomposition of \(B(z)\)  and testing whether they are
simultaneously triangularizable over \(\mathbb {C}\)  or not. For this (even if it is not necessary) one can form the Lie algebra \(\mathfrak {h}^{\prime }\)  generated by those matrices and test its solvability. It is not difficult to see that in fact \(\mathfrak {h}^{\prime } = \mathfrak {h}\)  (as defined above). Hence for systems with small coefficients there is no need to compute the differential Lie algebra \(\mathfrak {g}\) , in other words there is no need to compute a reduced form of the system, one has only to check the solvability of \(\mathfrak {h}^{\prime }\)  which is obtained directly from the coefficients of the system.
